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# How does this work?

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How does X in this equasion equals to 2 if 2-2=0 and you can't devide by zero? (x-1)/(x-2)-2/x=1/(x-2)

Guest Oct 8, 2017
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$$\frac{x-1}{x-2}-\frac{2}{x}\,=\,\frac{1}{x-2}$$

First let's get a common denominator.

$$\frac{(x)(x-1)}{(x)(x-2)}-\frac{2(x-2)}{(x)(x-2)}\,=\,\frac{1x}{(x)(x-2)}$$

We can multiply through by the denominator....however....you are right that x can't be 2 !

Also  x  can't be  0 . If we plug in  0  or  2  for  x  into the original equation, we will get an undefined result. Before we multiply through by the denominator, we must say that  x ≠ 0  and  x ≠ 2 .

$$(x)(x-1)-2(x-2) \,=\,1x$$         Distribute the  x  and the  -2 .

$$x^2-x-2x+4 \,=\,x$$           Subtract  x  from both sides and combine like terms.

$$x^2-4x+4 \,=\,0$$          Factor the left side and solve for  x  .

$$(x-2)^2=0 \\~\\ x-2=0 \\~\\ x=2$$

But, x cannot be 2 because it causes a zero in the denominator of the original problem.

So there is no solution.

hectictar  Oct 8, 2017
edited by hectictar  Oct 8, 2017

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